Quantum Hall effect in graphene: A functional determinant approach

0.17 MB·

by  C. G. Beneventano

International Workshop on Quantum Field Theory under the Influence of
External Conditions, QFEXT07, Leipzig, Germany
Quantum Hall effect in graphene: A functional
determinant approach
8
0 C G Beneventano and E M Santangelo
‡ §
0
Departamento deF´ısica,UniversidadNacionaldeLaPlata
2 InstitutodeF´ısicadeLaPlata,UNLP-CONICET
n C.C.67,1900LaPlata,Argentina
a E-mail: [email protected],
J [email protected]

0
2
Abstract.

We start the paper with a brief presentation of the main characteristics
]
of graphene, and of the Dirac theory of massless fermions in 2+1 dimensions
h
obtained as the associated low-momentum effective theory, in the absence of
t
- external fields. We then summarize the main steps needed to obtain the Hall
p
conductivity in the effective theory at finite temperature and density, with
e emphasisonitsdependenceonthephaseoftheDiracdeterminantselectedduring
h
the evaluation of the effective action. Finally, we discuss the behavior, under
[ gauge transformations, of the contribution due to the lowest Landau level, and

interpretgaugetransformationsasrotationsofthecorrespondingspinorsaround
2
themagneticfield.
v
8
2
9 PACSnumbers: 11.10.Wx,02.30.Sa,73.43.-f
4
.
0 1. Introduction
1
7
Graphene is a bidimensional array of carbon atoms, packed in a honeycomb crystal
0
structure. Actually, each layer of a graphene sample can be viewed as either an
:
v individual plane extracted from graphite, or else as an array of unrolled carbon
i
X nanotubes. Quite unexpectedly, in 2004, stable monolayer samples of such material
were obtained [1] and, in 2005, the Hall conductivity was measured in such samples,
r
a independently, by two groups [2]. More recently, a different behavior of the Hall
conductivity was reported [3] for bilayer samples. The main difference between the
behavior of the Hall conductivity of mono- and bilayer samples is in the height of the
jump around zero carrier density (or, equivalently, chemical potential).
Fromatheoreticalpointofview,themostremarkablefeatureofgrapheneisthat,
in a smallmomentum approximation,the chargecarriersorquasi–particlesbehaveas
two “flavors” (to account for the spin of the elementary constituents) of massless
relativistic Dirac particles in the two non–equivalent representations of the Clifford
algebra (corresponding to the two non–equivalentverticesin the first Brillouin zone),
with an effective “speed of light” about two orders of magnitude smaller than c [4].
‡ MemberofCONICET
§ MemberofCONICET
Quantum Hall effect in graphene 2
In[5],weshowedthatafieldtheorycalculationatfinitetemperatureanddensity,
based upon ζ function regularization of the Dirac determinant leads, in the zero
−
temperature limit, to a sequence of plateaux in the Hall conductivity consistent with
the measured ones, each time the chemical potential goes through a nonzero Landau
level. Moreover, it was shown in [6] that two of the three possible combinations of
phasesoftheDiracdeterminantinbothnonequivalentCliffordrepresentationspredict
abehavioraroundzerochemicalpotentialconsistentwiththeonesmeasuredinmono-
and bilayer graphene. For a complete presentation of other approaches to the study
of graphene see, for instance, [7], and references therein.
Thispaperpresents,insection2,abriefintroductiontothestructureofgraphene,
andtothederivationofthecontinuousDiraceffectivetheory,intheabsenceofexternal
fields. Section3containsareviewofourpreviousresultsonthesubject,withemphasis
ontheroleofthe phaseofthe determinantingivingrisetodifferentbehaviorsaround
zero chemical potential. In section 4, entirely new results are presented. In that
section,weallowforcomplexchemicalpotentials,andconcentrateonthecontribution
due to the lowestLandaulevel, inorderto study the invarianceofthe effective action
under large gauge transformations. By relating gauge transformations to rotations
in the plane, we analyze the effect of a 2π rotation for each of the three possible
combinations of phases in both representations, and identify, in the zero temperature
limit, the resulting geometrical or Berry’s phases. Our conclusions are presented in
section 5.
2. Structure of graphene. Effective continuous model
In this section, we sketch the main steps leading to the effective Dirac model for
graphene, in the absence of external fields. For more detailed presentations see, for
instance, [4].
The structure of the direct lattice for graphene is shown in figure 1. The direct
lattice is a superpositionoftwotriangularlattices,A andB.The generatorsoflattice
A are a = √3a(1, √3), and a = √3a(1,√3), where a is the lattice spacing. The
1 2 −2 2 2 2
vectors s = a(0, 1), s = a(√3,1) and s = a( √3,1) connect each site in the
1 − 2 2 2 3 −2 2
lattice A to its nearest neighbor sites in the lattice B. The tight binding Hamiltonian
can then be written as
3
H = t a (r)b(r+s )+b (r+s )a(r) ,
0 † i † i
−
rX∈ΛAXi=1(cid:2) (cid:3)
where t is the uniform hopping constant.
a(k) a(r)
In momentum space, with = e ikr , it reads
b(k) − · b(r)
(cid:26) (cid:27) rǫΛA (cid:26) (cid:27)
P
H = Φ(k)a (k)b(k)+Φ (k)b (k)a(k)
0 † ∗ †
k
X (cid:0) (cid:1)
3
with Φ(k)= t eik·si. After defining two-component spinors as
−
i=1
ψ(k) (a(k), b(kP))T ,ψ (k) a (k), b (k) , one gets
† † †
≡ ≡
0 (cid:0) Φ(k) (cid:1)
H = .
0 Φ (k) 0
∗
(cid:18) (cid:19)
Quantum Hall effect in graphene 3
Figure 1. Directlatticeforgraphene
H vanishes at the six corners of the first Brillouin zone. Among these, only two
0
are inequivalent, and can be chosen as
4π
k=K = ,0 ; Φ(K )=0.
± ± 3√3a ±
(cid:18) (cid:19)
When H is linearized around these two points , k=K +p, one obtains
0
±
0 p ip
Hk=K±+p =c p ip x∓0 y . (1)
x y
(cid:18) ± (cid:19)
This last expression shows that each Fermi point gives rise to an effective Dirac
theory, with an effective “velocity of light” c = 3ta, in one of the two inequivalent
2
representations of the gamma matrices. Thus, the total Hamiltonian can be taken as
the direct sum of both (equivalently, as the Dirac Hamiltonian in the reducible 4x4
representationoftheCliffordalgebra). Moreover,anoverallfactoroftwo(twofermion
species or “flavors”) must be included to take the spin of the original particles into
account.
3. The Hall conductivity from the interacting quantum field theory at
finite temperature and density
As shown in our previous work on the subject [5, 6], the Hall conductivity can
be determined by first evaluating the partition function (equivalently, the effective
action) for massless Dirac fermions at finite temperature and density, in two spacial
dimensions, in the presence of an external magnetic field perpendicular to the plane,
andthenperformingaboosttoareferenceframewithorthogonalelectricandmagnetic
fields. Inthissection,wesketchourmainresultsinthosereferences,withemphasison
the role played by the phase of the Dirac determinant, which appears when treating
the infinite tower of states associated to the lowest Landau level. We first consider a
single flavor,andoneofthe twononequivalentrepresentationsofthe Cliffordalgebra.
Inordertoconsidertheeffectsduetofinitetemperatureanddensity,westudythe
theory in Euclidean three-dimensional space, with a compact Euclidean “time” 0
≤
x β,whereβ = 1 (here,k istheBoltzmannconstantandT isthetemperature).
0 ≤ kBT B
We introduce the (real) chemical potential and the magnetic field through a minimal
coupling of the theory to an electromagnetic potential A = ( iµ,0,Bx ). Natural
units (c=~=1) will be used, unless otherwise stated. µ − e 1
In this scenario, the Euclidean effective action is given by log = log det(i/∂
Z −
eA/) , where the subindex AP indicates that antiperiodic boundary conditions must
AP
Quantum Hall effect in graphene 4
be imposed in the x direction, in order to ensure Fermi statistics. Now, this is a
0
formalexpression,whichwe willdefine throughazeta-functionregularization[8], i.e.,
d (i/∂ eA/) d ω s
S =log ζ(s, − AP)= − , (2)
eff
Z ≡−ds α −ds α
(cid:23)s=0 (cid:23)s=0Xω (cid:16) (cid:17)
where ω represents the eigenvalues of the Dirac operator acting on antiperiodic,
square-integrable functions, and α is a parameter introduced to render the zeta
function dimensionless (as expected on physical grounds, our final predictions will
be α-independent).
So, in order order to evaluate the partition function, we first determine the
eigenfunctions, andthe correspondingeigenvalues,ofthe Dirac operator. We propose
eiλlx0eikx2 ϕ (x ) π
Ψ (x ,x ,x )= k,l 1 λ =(2l+1) .
k,l 0 1 2 √2πβ χk,l(x1) l β
(cid:18) (cid:19)
Notethat,inthelastexpression,λ , l= ,..., aretheMatsubarafrequencies
l
−∞ ∞
adequate to the required antiperiodic conditions, while the continuous index k
represents an infinite degeneracy in the x direction.
2
The resulting spectrum has two pieces: An asymmetric piece, associated to the
lowest Landau level of the Hamiltonian:
π
ω =λ˜ , with λ˜ =(2l+1) +iµ and l = ,..., ,
l l l
β −∞ ∞
with corresponding eigenfunctions
ψk,l(x1)= eπB 14 e−e2B(x1−ekB)2 , (3)
0 !
(cid:0) (cid:1)
and a symmetric piece
ω = λ˜2+2neB with n=1,..., l = ,..., ,
l,n ± l ∞ −∞ ∞
correspondingtoeigenfunqctionswithbothcomponentsdifferentfromzero. Inallcases,
the degeneracy per unit area is given by the well known Landau factor, ∆ = eB.
L 2π
Theasymmetricpartofthespectrumisquiteparticular. Inthefirstplace,asseen
from (3), the corresponding eigenfunction is an eigenfunction of the Pauli matrix σ ,
3
with eigenvalue +1. The eigenfunction with the opposite “chirality” was eliminated
by the square integrability condition in x . Moreover, in the other nonequivalent
1
representation, this part of the spectrum appears with the opposite sign. Such
transformation is equivalent to µ µ. This is nothing but charge conjugation
→ −
(which,forrealµ,isalsoequivalenttoaparitytransformation[9]). Aswewilldiscuss
in what follows, this part of the spectrum is the one which requires the consideration
of a phase of the determinant when evaluating the effective action.
Before going to such evaluation, it is interesting to note the invariance of the
whole spectrum under µ µ+ 2ikπ. This invariance is a natural one, since such
→ β
transformationspreservetheantiperiodicityoftheeigenfunctionsand,thus,theDirac
statistics. They are nothing but the so-called large gauge transformations. The
invariance of the effective action under such transformations is also required for
topological reasons. We will discuss this point in more detail in section 4.
As is clear from (2), in evaluating the effective action, one must perform the
analytic extension of the contributions to the zeta function coming from both pieces
of the spectrum,
s
∞ π µ −
ζ (s,µ)=∆ (2l+1) +i ,
1 L
αβ α
l= (cid:20) (cid:21)
X−∞
Quantum Hall effect in graphene 5
and
s
∞ 2neB π µ 2 −2
ζ (s,µ,eB)=(1+( 1) s)∆ + (2l+1) +i .
2 − − L nX=1 " α2 (cid:18) αβ α(cid:19) #
l=
−∞
The analytic extension of ζ (s,µ,eB) is quite standard, and it relies mainly on
2
performingaMellintransformandmakinguseoftheinversionpropertiesoftheJacobi
theta functions. A detailed presentation can be found in [5]. The final result for the
contribution to the effective action coming from this piece is (always consideringonly
one representation of the gamma matrices and one fermion species)
SII =∆ β√2eBζ 1 +∆ ∞ log 1+e (√2neB µ)β 1+e (√2neB+µ)β . (4)
eff L R −2 L − − −
(cid:18) (cid:19) nX=1 h(cid:16) (cid:17)(cid:16) (cid:17)i
The other nonequivalent representation of the Clifford algebra gives rise to an
identical contribution, since this part of the spectrum is the same in both irreducible
representations.
Assaidbefore,theextensionofζ (s,µ,eB)requiresacarefulconsiderationofthe
1
phase of the determinant. In fact, ζ can be written as
1
s s s
2π − ∞ 1 µβ − ∞ 1 µβ −
ζ (s,µ)=∆ (l+ )+i + (l+ ) i , (5)
1 L
(cid:18)αβ(cid:19) "l=0(cid:20) 2 2π(cid:21) l=0(cid:20)−(cid:18) 2 − 2π(cid:19)(cid:21) #
X X
andthedefinitionoftheoverallminussigninthesecondsumdependsontheselection
of the cut in the complex plane of eigenvalues. As discussed in detail in [6], the usual
prescriptionistochoosethecutsuchthatonedoesnotgothroughvanishingarguments
when continuously transforming eigenvalues with positive real part into eigenvalues
with negative real part [10]. This prescription then gives rise to what will be called
inthe followingthe standardphase ofthe determinant(characterizedfromnowonby
κ= 1). One could certainly choose the opposite prescription, which we will call the
−
nonstandard phase (κ = +1). Once one of the phases is selected, the contribution of
ζ tothe effectiveactioncanbeevaluatedbymakinguseofthe well-knownproperties
1
of the Hurwitz zeta function, to obtain
µβ µβ
SI (κ)=∆ log 2cosh( ) +κ| | .
eff L 2 2
(cid:26) (cid:20) (cid:21) (cid:27)
When this last contribution is added to the one in (4), one gets for the effective
action
µβ µβ 1
S (κ)=∆ log 2cosh( ) +κ| | +β√2eBζ
eff L R
2 2 −2
(cid:26) (cid:20) (cid:21) (cid:18) (cid:19)
+ ∞ log 1+e−(√2neB−µ)β 1+e−(√2neB+µ)β .
)
nX=1 h(cid:16) (cid:17)(cid:16) (cid:17)i
From this last expression, the finite-temperature charge density can be obtained
as j (κ) = e d log . In the zero-temperature (β ), and recovering physical
0 −β dµ Z → ∞
units, it reduces to
(n+ 1+κ)ce2B
j0(2ec2~Bn<µ2 <2eBc2~(n+1))= − 2 sign(µ),
h
Quantum Hall effect in graphene 6
3 2
2
2
1 1 1
-3-2-1 1 2 3 -3-2-1 1 2 3 -3-2-1 1 2 3
-1 -1 -1
-2
-2
-3 -2
Figure 2. Hall conductivity for different selections of the phase of the
determinant. Left to right: K = 1, K = 2, K = 0. In all cases, the horizontal
axisrepresentsνC =sgn(µ)µ2/2eB~c2 andtheverticalone,σxyh/4e2.
where n=[ µ2 ], and [x] is the integer part of x.
2eB~c
In orderto obtainthe Hallconductivity,one must performaboost to a reference
frame with crossed electric and magnetic fields. The final contribution to the Hall
conductivity from each fermion species and one irreducible representation is given by
(n+1+κ)e2
[6] σxy = − h2 sign(µ).
Now, the phases of the determinant in both irreducible representations can be
selectedwiththesameorwithoppositecriteria. Whenthis istakenintoaccount,and
anoverallfactorof2isincluded,totakebothfermionspeciesintoaccount,onobtains
for the total zero-temperature Hall conductivity
4(n+ K)e2
σ = − 2 sign(µ),
xy
h
where K =0 corresponds to selecting the standard phase of the determinant in both
irreducible representations, K = 1 corresponds to choosing opposite criteria for the
phases,andK =2,tochoosingbothphasesinthenonstandardway. Thedependence
of the Hall conductivity on the classical filling factor (ν ) is presented in figure 2, for
C
the three values of K. From that figure, it is clear that the behavior of monolayer
graphene, as presented in [2], corresponds to K =1, i.e., to choosing opposite phases
ofthedeterminantinbothrepresentations(or,equivalently,ignoringthephaseinboth
representations, as done in [11]). In fact, in this case the (rescaled) Hall conductivity
shows a jump of height 1 for ν = 0, and further jumps of the same magnitude for
C
ν = 1, 2,.... Inturn,thebehaviorofbilayergraphene,asreportedin[3]isexactly
C
± ±
reproduced by K =2 (nonstandard selection of the phase in both representations).
4. Invariance under large gauge transformations. Interpretation in terms
of rotations
To analyze the physical meaning of the invariance of the effective action under large
gauge transformations in this context, we go back to the zeta function associated to
the asymmetric part of the spectrum, for two fermion species and one representation,
this time allowing for an imaginary part in the chemical potential, µ˜ =µ+iγ, while
alwayskeepingµ=0. Inthiscase,onemustbecarefulwhensplittingtheinfinitesum
6
asin(5)(detailofthe calculationswillappearin[12]). Infact,suchsplitting mustbe
different for different γ-ranges, to make sure that all the eigenvalues in each infinite
sum have a real part with the same sign, which is crucial in defining the phase. For
Quantum Hall effect in graphene 7
example, for 1 < γβ < 1, one has
−2 2π 2
SeIff(−12 < γ2πβ < 12)= −2∆L dds(cid:23)s=0(l∞=0[(2l+1)π/β+iµ−γ]−s
X
+ ∞ e−isθ (2l+1)π/β+i(µ+iγ)e−iθ −s .
)
l=0
X (cid:2) (cid:3)
Now,the values ofθ suchthatthe secondterminthe RHSdoes vanisharethoseones
for which, simultaneously, (2l+1)π/β+µsinθ γcosθ = 0 =µcosθ+γsinθ. As
−
before,weconsiderheretwodifferentdefinitionsofthephaseofthedeterminant,which
correspond to the standard definition for the phase κ = 1, and to the nonstandard
−
oneκ=+1. Witheachoneoftheseprescriptions,the contributionofthe asymmetric
spectrum to the effective a action is given by
1 γβ 1 (κ+1)β
SI ( < < )=2∆ sgnµ(µ+iγ)
eff −2 2π 2 L 2
(cid:26)
+log e−β2(µ+iγ)(1+sgnµ)+eβ2(µ+iγ)(1−sgnµ) . (6)
(cid:16) (cid:17)o
Things are entirely different for γβ = 1. In this case, one mode in the infinite
2π ±2
sum defining the zeta function has a vanishing real part. A careful treatment shows
that, at such points, SI is discontinuous. For instance, SI (γβ = +1) coincides
eff eff 2π 2
with limγ2πβ→21− of (6). An equally carefully treatment of the case γ2πβ = −21 shows
that SI (γβ = 1)=SI (γβ = 1). This analysis can be extended to other ranges
eff 2π −2 eff 2π 2
of variation of γβ, to obtain, with k = ,..., ,
2π −∞ ∞
1 γβ 1 (κ+1)β 2kπ
SI ((k )< (k+ ))=2∆ sgnµ[µ+i(γ )]
eff − 2 2π ≤ 2 L 2 − β
(cid:26)
+ log e−β2(µ+i(γ−2kβπ))(1+sgnµ)+eβ2(µ+i(γ−2kβπ))(1−sgnµ) . (7)
(cid:16) (cid:17)o
This expression shows that the contribution to the effective action of the
nonsymmetric part of the spectrum, in this representation of the gamma matrices, is
invariantunderlargegaugetransformations,nomatterwhichphaseofthedeterminant
is selected. As already said, such transformations must constitute an invariance. In
fact,anincreaseofiγinthechemicalpotentialcorrespondstothemultiplicationofthe
eigenfunctions (3) with a phase, i.e., ψk,l(x)→eiγx0ψk,l(x). So, an increase iγ = 2βiπ
is a pure gauge transformation which, moreover,preserves the antiperiodicity in x .
0
Due to the fact that these eigenfunctions are eigenfunctions of σ such that
3
σ ψ (x) = ψ (x), one can equivalently write gauge transformations in the form
3 n=o n=o
ψk,l(x) eiσ232γx0ψk,l(x). This last expression shows that, as x0 grows from 0 to β,
spinors →are rotated by 2γβ, since σ3 is the generator of rotations in the plane x x .
2 1 2
In particular, γ = 2π corresponds to a 4π-rotation around the magnetic field. Note
β
that not only the partition function, but the Abelian Chern-Simons term (and, thus,
the effective action) is invariant under large gauge transformations. On the other
hand,γ = π correspondstoa2π-rotation. Atfinitetemperature,suchtransformation
β
changesthestatisticstoabosonicone. Forκ=+1,italsogivesrisetoanoverallphase
ofπperunitdegeneracyinthepartitionfunction. Suchphaseisthecontributionwhich
survives in the zero temperature limit. The three possible combinations of phases of
the determinant then give a total phase per unit degeneracy in the partition function
Quantum Hall effect in graphene 8
of π (K = 1, which reproduces the behavior of the Hall conductivity for monolayer
graphene), or 0 (both for K =0 and K =2, this last reproducing the behavior of the
Hall conductivity for bilayer graphene). At his point, it is interesting to note that, in
ordertohaveazero-temperaturepartitionfunctioninvariantunderrotationsof2πfor
monolayer graphene, the reduced flux through a unit cell of area Ω must be given by
Φ =Ω∆ =N,withN apositiveinteger. Thisispreciselytheconditionforphysical
Φ0 L
states to transform as unidimensional ray representations of the magnetic translation
group [9].
5. Conclusions
The first conclusion, as already stated in [6], is that two of the three possible
combinationsofphasesgivebehaviorsoftheHallconductivitywhichcoincidewiththe
ones measuredinmono- andbilayergraphene. Inthe caseofbilayergraphene[3],the
(rescaled)Hallconductivitypresentsajumpofheight2forν =0,andfurtherjumps
C
of height 1. The main point here concerns the positions of these subsequent jumps.
As a matter of fact, according to figure 1.b in [3], these subsequent jumps appear
for ν = 1, 2,..., which is exactly the behavior predicted, in our calculation, for
C
± ±
K =+2. However,thesamereferenceinterpretstheHallbehaviorofbilayergraphene
throughatheoreticalpredictionfirstmade in [13]which, asdiscussedin[14], predicts
a plateaux of larger width. Our calculation, instead, completely coincides with the
measured behavior of the plateaux, both in height and width.
An entirely new conclusion is that, in each representation, the effective action
per unit degeneracy is invariant under large gauge transformations, with any of the
two possible selections of phase. As a result, the invariance persists no matter which
of the three possible combinations of phases is selected. Moreover, each of the two
selections of phase in each representation corresponds to a different geometric phase
undertherotationofspinorsalongaclosedpatharoundthemagneticfield(κ= 1: no
−
geometricphase; κ=+1: geometricphase ofπ). So, differentvalues ofK correspond
to different total geometric phases per unit degeneracy, to be compared with the
Berry phases studied, for instance, in [14]. Finally, by taking γβ = π, we note that,
3
for Φ =1,thesethreevaluesofK alsocorrespondtothethreenonequivalentunitary
Φ0
representationsofthegeneratorofthecyclicgroupC ,whichistherelevantsymmetry
3
in the case of free graphene.
To the best of our knowledge, the relation between the phase of the fermionic
determinant and Berry’s phase hadn’t been noticed before. This point, as well as the
connection with the magnetic translationgroup will be studied in more detail in [12].
Acknowledgments
E.M.S. thanks the organizers of QFEXT07 for the perfect organization, and for the
nice atmosphere enjoyed during the event. This work was partially supported by
Universidad Nacional de La Plata (Proyecto 11/X381)and CONICET (PIP 6160).
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